Preimage entropy for random dynamical systems

In [6], Cheng and Newhouse introduced and studied the new
invariants - preimage entropies for deterministic dynamical systems.
In this paper, the analogous notions, measure-theoretic preimage
entropy and topological preimage entropy, are formulated for random
dynamical systems. Analogues of many known results for entropies,
such as the Shannon-McMillan-Breiman Theorem, the Kolmogorov-Sinai
Theorem, the Abromov-Rokhlin formula and the power rule, are
obtained for preimage entropies. In particular, a variational
principle is given.